Energy efficiency of compressed spectrum sensing in wideband cognitive radio networks
 Qi Zhao^{1}Email author,
 Zhijie Wu^{2} and
 Xiaochun Li^{1}
https://doi.org/10.1186/s1363801605819
© Zhao et al. 2016
Received: 16 April 2015
Accepted: 3 March 2016
Published: 15 March 2016
Abstract
In cognitive radio networks, wideband spectrum sensing (WSS) has been advocated as an effective approach to increase the spectrum access opportunity of the secondary users. On the other hand, because of the increase of sampling rate and sensing time duration cost by the analogtodigital converter (ADC), energy saving has been a significant problem for WSS. In this paper, taking advantage of the frequencydomain sparsity of the wideband spectrum, a WSS scheme combining compressed sensing and multiband joint detection technique is proposed to reduce the energy consumption. Based on extensive analysis and simulation, we identify the sparsity order of the wideband spectrum, the received signaltonoise ratio (SNR) of the primary signal, and the compression rate employed in sampling as three key factors that affect the sensing performance. In particular, we derive a closedform analytical model of the scheme. Based on these observations, the energy efficiency, defined as the ratio of the spectrum access opportunity to the energy consumption, is maximized through the optimization process of the compression rate under the sensing performance constraints. We also indicate the uniqueness of the optimal compression rate for maximizing the energy efficiency. Numerical and simulation results show that our proposed scheme is more energy efficient if the wideband spectrum is sparser in the frequency domain.
Keywords
1 Introduction
In cognitive radio networks (CRNs), the secondary users (SUs) can opportunistically access the spectrum bands unoccupied by the primary users (PUs). As a key technology in CRNs, spectrum sensing is widely used by SUs to periodically detect the spectrum bands. It is well known that the greater the bandwidth of the spectrum being detected, the more opportunity for SUs to access. Therefore, wideband spectrum sensing (WSS) [1, 2] is attracting attention as an effective approach for detecting continuous multiple bands simultaneously. Since WSS can be easily performed in local sensing, i.e., it can be implemented by any SU independently, the communication overhead for network coordination can be greatly reduced. Hence, WSS is suitable for largescale CRNs with quite a lot of SUs.
On the other hand, the increasing energy consumption has been a significant problem for WSS [1, 3]. The energy consumption in the spectrum sensing is mainly caused by the analogtodigital converter (ADC), which is proportional to the sensing time duration and the sampling rate [1]. However, because of the wide range of the spectrum bandwidth for WSS, long sensing time duration and high sampling rate are always needed by the ADC. There are two kinds of the traditional WSS schemes, the sequential sensing scheme [1, 3] and the parallel multiband detection scheme [4]. Originally, constrained by the low sampling rate of ADCs, SU divides the wideband spectrum into multiple subbands and then detects them successively, namely the sequential sensing scheme. As the highspeed sampling technique develops, SU can directly sample the wideband spectrum using a highspeed ADC and then detect all the subbands simultaneously, namely the parallel multiband detection scheme. Compared with the sequential sensing scheme, the structure of the parallel multiband detection scheme is more convenient for WSS. Furthermore, the parallel multiband detection scheme is more energy efficient by improving the sampling rate to shorten the sensing time duration.
In practice, the wideband spectrum is divided into several nonoverlapping narrowband channels assigned to different PUs, and the emergence of different PUs’ signal is entirely independent. Therefore, only a minority of channels are occupied by PUs at the same time, namely that the wideband spectrum is sparser in the frequency range. Taking advantage of such sparsity of wideband spectrum, compressed sensing (CS) has recently been proposed to reduce the sampling rate below the Nyquist rate [5–11]. For this reason, the CSbased spectrum sensing methods have been proposed as an efficient approach for energy saving in WSS.
Tian and Giannakis introduced two kinds of compressed sensing methods (called multistep and onestep) to recover the signal by wavelet approach [5] and then developed the signal recovery idea and performed the distributed compressive spectrum sensing in the cooperative multihop CRNs [6]. However, for spectrum sensing and detection, accurately reconstructing the original signal is unnecessary. Following this idea, Polo et al. presented a compressive wideband spectrum sensing scheme by sampling the analog signal using analogtoinformation convert and then focused on the performance of edge spectrum and power spectrum density recovery [7]. The authors in [8, 9] also suggested to only reconstruct the signal’s power spectrum, in order to optimize the sensing performance at the expected probabilities of detection and false alarm. As well known, the final purpose of spectrum sensing is to find the spectrum access opportunities for SUs [1, 5]. Based on this point, all the aforementioned methods, aiming at the signal reconstruction performance in [5, 6] or the sensing performance in [7–9], give no considerations to the optimization problem of the spectrum access opportunity. In this paper, we have a further study on this problem, combining the challenge of energy consumption in WSS. We propose a reconstruction structure aiming at optimizing the energy efficiency, which is defined as the ratio of SU’s spectrum access opportunity to the energy consumption.
The proposed design includes (i) a sensing scheme combining the compressed sensing and the multiband joint detection, (ii) the establishment of an analytical model of the scheme, and (iii) the optimization of the compression rate for energy efficiency maximization under the sensing performance constraints. The contributions of this paper can be generalized as follows.
In comparison with the traditional WSS schemes, the proposed scheme reduces the sampling rate at the subNyquist rate and takes a shorter sensing time duration, both of which are greatly favorable to energy saving. Numerical and simulation results show that the performance of energy saving is better if the wideband spectrum is sparser in the frequency domain.
Although the compressed sensing has been suggested to reduce the sampling rate in WSS, the energy efficiency and the optimization of compression rate have never been considered. However, our study indicates the uniqueness of the optimal compression rate for maximizing the energy efficiency.
Furthermore, lots of relevant studies are developed under the ideal conditions. As in [10, 11], the authors deduced the minimum required number of measurements for sparsity estimation and signal reconstruction, respectively, but in the noisefree case. Based on extensive analysis and simulation, we identify the sparsity order, the received signaltonoise ratio (SNR), and the compression rate as three key factors when performing our proposed scheme and then quantify these contributions by mathematical approximation to support the theoretical derivation.
In addition, a joint SNR and sparsity order estimation, and the conditions for implementing the compressed sensing, are suggested to give consideration to the practical application of the proposed scheme.
The remainder of this paper is organized as follows. Section 2 describes the signal and network model. Section 3 presents the composition of the proposed scheme, formulates the energy efficiency maximization problem, and derives the optimal compression rate. Section 4 shows the performance analysis and simulations. Finally, Section 5 concludes the paper.
2 Network mode
Consider a wideband spectrum that is divided into K nonoverlapping subbands. During a constant sensing time duration, we model the detection problem on the subband k as one choosing between a hypothesis \( {H}_{0,\;k} \), which represents the absence of the primary signals, and an alternative hypothesis \( {H}_{1,\;k} \), which represents the presence of the primary signals, k ∈ {1, 2, …, K}. Moreover, we have \( \Pr \left({H}_{0,\;k}\right)+ \Pr \left({H}_{1,\;k}\right)=1 \), where Pr(⋅) is the probability of occurrence. Without loss of generality, we assume that \( \Pr \left({H}_{0,\;k}\right)= \Pr \left({H}_0\right) \), and \( \Pr \left({H}_{1,\;k}\right)= \Pr \left({H}_1\right) \), ∀ k. Let S _{ k }(t) and H _{ k }(t) be the primary signal and the coefficient of the subband k at any given time t, respectively. Then, define X(t) = {X _{1}(t), X _{2}(t), ⋯, X _{ K }(t)}, where X _{ k }(t) = H _{ k }(t)S _{ k }(t). Thus, the received signals at SUs can be represented as R(t) = X(t) ⋅ 1(k, t) + V(t), where V(t) is the additive white Gaussian noise and 1(k, t) is a K × 1 column vector in which the kth element is equal to 1 in the hypothesis \( {H}_{1,\;k} \) and is equal to 0 in the hypothesis \( {H}_{0,\;k} \). In addition, we define the received SNR on the subband k as γ _{ k } = p _{ k }/p _{0}, where p _{ k } is the received power of the primary signals and p _{0} the noise power.
3 Proposed wideband spectrum sensing scheme
3.1 Scheme description
If C _{ 1 } is satisfied, i.e., the spectrum is sparse and all the elements of γ = {γ _{ k }, ∀ k} are above the lower limit \( \overline{\gamma} \), in A _{ 1 }, R(t) is compressed sampled at the optimal subNyquist sampling rate, given by A _{ 3 }, for the energy efficiency maximization, and then reconstructed through the orthogonal matching pursuit (OMP) method [4] as \( \widehat{\boldsymbol{R}}=\left\{\widehat{R}(n)\right\} \), n = {1, 2, …, N}. Otherwise, a ⋅ exp(b ⋅ δ) + c ⋅ exp(d ⋅ δ) is sampled at the Nyquist rate as R = {R(n)}, n = {1, 2, …, N}. Next, \( \widehat{\boldsymbol{R}} \) (or R) is used by A _{ 2 } for the parallel multiband detection. Meanwhile, the summary statistic of each subband k, \( {\widehat{T}}_k \) (or T _{ k }), is used by B _{ 1 } to estimate the sparsity order and γ.
where f _{ s } is the sampling rate and τ _{ s } is the sampling time duration; the units of E, f _{ s }, and τ _{ s } are mW, Mbps, and s, respectively. It can be observed that when f _{ s } is in a relatively low regime, i.e., f _{ s } ≤ 65MHz, E is mainly dependent on τ _{ s }. This indicates that for a given N, where N = f _{ s } τ _{ s }, increasing f _{ s } (as a means of shortening τ _{ s }) is efficient to save energy. Compared with the sequential sensing scheme, the parallel detection structure of A _{ 2 } is more suitable for sampling the wideband spectrum at a high f _{ s }. However, when f _{ s } > 65 MHz, the energy consumption caused by the increase of f _{ s } is apparent. To overcome this, A _{ 1 } is applied to reduce f _{ s } to a level of less than the Nyquist rate for further energy saving.
3.2 Compressed sensing (A _{ 1 })
In most cases, since X(t) does not occupy all the K subbands, it may be sparse in the frequency domain. According to the compressed sensing theory [13], if X(t) is truly sparse, it can be sampled at the subNyquist rate and then reconstructed with slight errors. Denote the sparsity order of X(t) as \( \rho =\frac{\mathrm{the}\kern0.5em \mathrm{number}\kern0.5em \mathrm{of}\kern0.5em \mathrm{occupied}\kern0.5em \mathrm{s}\mathrm{u}\mathrm{b}\hbox{} \mathrm{bands}}{K} \) and the compression rate for sampling as δ = f _{cs}/f _{ N }, δ ∈ (0, 1), where f _{cs} is the compressed sampling rate and f _{ N } is the Nyquist rate. It is well known that δ must be larger than ρ in order to reconstruct the sparse signals. Moreover, the reconstruction performance is improved with the increase of δ.

Choose an N × N discrete Fourier transform (DFT) matrix as the sparse representation basis Ψ, then map R into the sparse domain as Θ = Ψ ^{T} R ^{T}.

Choose a V × N uniform Gaussian random sampling matrix Φ, where V = ⌈δN⌉ (⌈x⌉ is the smallest integer not less than x), then calculate the measurement matrix as W = ΦΨ ^{T} R ^{T}.

Initialization. Denote the iteration index as i = 0, the error vector as r ^{(0)} = W, and the expected nonnegative error vector as \( \overline{\boldsymbol{r}} \). Let \( {\widehat{\boldsymbol{R}}}^{(0)}=0 \), Ω ^{(0)} = Ø, where 0 is a 1 × N vector consisting of zero elements.

Step 1: Calculate the inner products of the error vector and the columns of Λ, g ^{(i)} = Λ ^{T} r ^{(i − 1)}.

Step 2: Find the maximum element of g ^{(i)}, then get its column index as \( j=\underset{n\in \left\{1,2,\cdots, N\right\}}{ \arg \max}\left{\boldsymbol{g}}^{(i)}(n)\right \).

Step 3: Update Ω ^{(i)} as Ω ^{(i − 1)} ∪ {Λ(:, j)}, where Λ(:, j) represents all the elements of the jth column of Λ.

Step 4: Calculate the approximate solution of R through the least square method, \( {\widehat{\boldsymbol{R}}}^{(i)}={\left({\left({\boldsymbol{\varOmega}}^{(i)}\right)}^{\mathrm{T}}{\boldsymbol{\varOmega}}^{(i)}\right)}^{1}{\left({\boldsymbol{\varOmega}}^{(i)}\right)}^{\mathrm{T}}\boldsymbol{W} \).

Step 5: Update r ^{(i)} as \( \boldsymbol{W}\boldsymbol{\varLambda} {\widehat{\boldsymbol{R}}}^{(i)} \).

Step 6: If \( \left{\boldsymbol{r}}^{(i)}\right\le \overline{\boldsymbol{r}} \) or i = V, output \( \widehat{\boldsymbol{R}}={\widehat{\boldsymbol{R}}}^{(i)} \) and stop. Otherwise, go to step 1.
So far, the compressed sampling and the signal reconstruction submodules have been present, and the simulation model of the compressed sensing can be established as a result.
3.3 Multiband joint detection (A _{ 2 })

Calculate the discrete frequency response of R (denoted as Y = {Y(n)}, n = {1, 2, …, N}) through an Npoint fast Fourier transform (FFT).

For each subband k, calculate summary statistic T _{ k } as the sum of the received signal energy over an interval of M samples, i.e., \( {T}_k={\displaystyle {\sum}_{m=1}^M{\left{Y}_k(m)\right}^2} \), where M = N/2K and n = (k − 1)M + m.

Compare T _{ k } with the threshold λ _{ k } to make a decision whether the primary signal is present or not.
3.4 Impact of sparsity order, compression rate, and SNR
Now, we consider the multiband joint detection when the compressed sensing module A _{ 1 } is valid. The spectrum can be reconstructed through the OMP method but at the expense of causing recovery errors. The performance of signal reconstruction depends on the sparsity order and the compression rate. Thus, it is foreseeable that the two factors should also affect the detection accuracy. On the other hand, since the energy detector used at each branch of A _{ 2 } cannot extract the primary signal from the noise, the received SNR is the most important factor for signal detection. In what follows, we thus investigate the impact of sparsity order, compression rate, and SNR on the sensing performance of the proposed scheme.
3.5 Analysis model
Experimental data of the coefficients, ρ = 0.2, γ _{ k } = 9 dB
δ  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9 

α _{1}  0.96  1.02  1.03  1.02  1.01  1.01  1.01  1.00 
α _{0}  10.93  4.06  1.53  1.05  0.83  0.71  0.61  0.57 
β _{1}  11.56  5.73  2.95  2.29  1.96  1.72  1.59  1.52 
β _{0}  48.79  17.42  5.29  3.48  2.65  2.30  1.97  1.80 
Fitting results of the coefficients, ρ = 0.2, γ _{ k } = 9 dB
a  b  c  d  

α _{0}  89.04  −10.59  0.461  0.277 
β _{1}  63.67  −9.23  1.73  −0.12 
β _{0}  427.7  −10.81  0.77  1.01 
3.6 Energy efficiency maximization (A _{ 3 })
The constraint (10) ensures that the primary transmission is not interrupted by the maximization process. The constraints (11) and (12) give an exact range of the compression rate.
then we have \( \underset{\delta \to \rho,\ \rho \to 0}{ \lim }{\eta}_k^{\prime }>0 \). On the other hand, when δ → 1, υ is close to be a constant being equal to \( \sqrt{2{\gamma}_k+1}\kern0.5em {Q}^{1}\left({\overline{P}}_d\right)+\sqrt{M/2}\left({\gamma}_k+1\right) \); thus, \( \underset{\delta \to 1}{ \lim }{\upsilon}^{\prime}\approx 0 \). Moreover, there is υ > 0 when P ^{–} _{ d } ≥ 0.5. Then, we have \( \underset{\delta \to 1}{ \lim }A\approx 0 \) and \( \underset{\delta \to 1}{ \lim }B\approx {E}^{\prime }>0 \); thus, \( \underset{\delta \to 1}{ \lim }{\eta}_k^{\prime }<0 \). As a result, there is a zero point of \( {\eta}_k^{\prime } \) within the interval of δ ∈ (0, 1), so the convexity of η _{ k } is proved. Considering all the K independent subbands, the maximum η must also be unique within the range of δ, and it can be easily found through the onedimensional exhaustive search. In practical application, we can replace the module A _{ 3 } by an optimal compression rate matrix indexed by the sparsity order ρ and the SNR vector γ, which can be stored in the computer’s memory. When the estimation of ρ and γ is finished, the compression rate δ can be chosen by searching through the matrix. This makes the energy consumed by A _{ 3 } be neglected compared with that of ADC.
3.7 SNR and sparsity order estimation (B _{ 1 })

Initialization: Let l = 1 and the set of accepted \( {H}_{k,l} \) for subband k as \( {\boldsymbol{H}}_k=\left\{{H}_{k,1},{H}_{k,2},\cdots, {H}_{k,L}\right\} \).

Step 1: At the signal level l, sort p(:, l) in ascending order of p values, p ^{(1)} ≤ p ^{(2)} ≤ ⋯ ≤ p ^{(K)}, where p(:, l) represents all the elements of the lth column of p.

Step 2: Find the maximum index of p ^{(j)} satisfying the condition that \( {p}^{(j)}\le \frac{j}{L}\varepsilon \), j ∈ {1, 2, ⋯, K}, and denote it as J.

Step 3: Reject all the \( {H}_{k,l} \) if k ∈ [1, J], then eliminate them from H _{ k }.

Step 4: If l = L, break the loop. Otherwise, l = l + 1, then go to step 1.

Step 5: If H _{ k } ≠ Ø, choose the maximum l from H _{ k } as the SNR level of subband k. Otherwise, classify it into the noise level.
Considering all the K subbands, the estimation of γ can be obtained. Furthermore, after the SNR classification, ρ can be estimated by \( \frac{\mathrm{the}\kern0.5em \mathrm{number}\kern0.5em \mathrm{of}\kern0.5em \mathrm{S}\mathrm{N}\mathrm{R}\kern0.5em \mathrm{classified}\kern0.5em \mathrm{into}\kern0.5em \mathrm{signal}\kern0.5em \mathrm{levels}}{K} \).
3.8 Compressed sampling decision (C _{ 1 })
when ρ = 0.5 and δ = 1, to ensure M is not less than the minimum number of required samples to achieve the pair of target probabilities \( \left({\overline{P}}_d,{\overline{P}}_f\right) \).
4 Performance analysis and simulations
In this section, the performance of energy efficiency maximization and energy saving is demonstrated by simulations. We consider a sparse wideband spectrum contains 10 TV bands each with 6MHz bandwidth. Assume the occurrence probabilities as \( \Pr \left({H}_1\right)=q \), \( \Pr \left({H}_0\right)=1q \), and q ∈ (0, 1), the number of signal levels as L = 10, and the received SNRs of the primary signal on each subband as γ = {0, 1, …, L − 1} (dB). Moreover, assume that \( {\overline{P}}_d=0.9 \), \( {\overline{P}}_f=0.1 \), N = 512, ε = 0.95, S = 20, and p _{0} = − 95.2 dBm.
5 Conclusions
Taking advantage of the frequencydomain sparsity of the wideband spectrum, a new wideband spectrum sensing scheme has been proposed to save energy through reducing both the sampling rate and sensing time duration. The sparsity order, the received signaltonoise ratio, and the compression rate are identified as three key factors that affect the sensing performance. Moreover, under the sensing performance constraints, the uniqueness of the maximum energy efficiency within the range of the compression rate has been proved. It is indicated that the sparser the wideband spectrum, the higher the energy efficiency and the lower the energy consumption. All these advantages have been corroborated by simulations.
Declarations
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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